This is one of the most stunningly cool things I've learned about the prime numbers: it's trivial to construct infinitely large gaps between them.
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To be sure, we want to be careful about the sense in which we're using "infinitely large" here. We don't mean that the gap itself is boundless. But we do mean infinitely large in the same sense in which some limits diverge to infinity. There is no upper boundary on the size of the gaps we can create.
It works like this: take any size gap you want. For example, let's say 6. If you want to construct a gap of at least six, we'll need to make a string of at least five integers that are *guaranteed* to contain no primes. To do that, use the factorial of the size gap we're creating. That factorial is guaranteed to be a multiple of all the numbers 1 up to that number itself (in our case, 1 through 6). And so anything 2 larger than that number must be even. Anything 3 larger must be a multiple of 3. Anything 4 larger must be a multiple of 4, and so on. In this case, that means the numbers 6! + 2 (which is equal to 722) through 6! + 6 (726) *must* be composite. So there you have it! We have constructed five integers in a row that must be composite, meaning at minimum, even if 6! + 1 and 6! + 7 were both prime, our prime gap would be at least 6 (the difference between those two numbers).
Now, in this case, the prime gap is in fact 8, between the primes 719 and 727 (meaning neither 720 nor 721 were prime). This shows us that we cannot precisely construct prime gaps, but we can guarantee that they are at least so large.
This is also a good reason to believe that the primes get less and less numerous the further down the number line you travel. Not only do the factorials themselves have to be surrounded by a certain number of composite numbers, their multiples share the same property. So even if there were no other reason to believe the prime numbers got less common the further down the number line you go, the fact that the factorials and their multiples become more and more common down the number line would still have you expect the primes must grow less dense.
#primenumbers #numbertheory #infinity
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